Distributed Robust Output Regulation of Heterogeneous Uncertain Linear Agents by Adaptive Internal Model Principle

Distributed Robust Output Regulation of Heterogeneous Uncertain Linear Agents by Adaptive Internal Model Principle

Satoshi Kawamura and Kai Cai This work was supported by the Research and Development of Innovative Network Technologies to Create the Future of National Institute of Information and Communications Technology (NICT) of Japan.Satoshi Kawamura and Kai Cai are with Department of Electrical and Information Engineering, Osaka City University, Japan. kawamura@c.info.eng.osaka-cu.ac.jp, kai.cai@eng.osaka-cu.ac.jp
Abstract

We study a multi-agent output regulation problem, where not all agents have access to the exosystem’s dynamics. We propose a distributed controller that solves the problem for linear, heterogeneous, and uncertain agent dynamics as well as time-varying directed networks. The distributed controller consists of two parts: (1) an exosystem generator that creates a local copy of the exosystem dynamics by using consensus protocols, and (2) a dynamic compensator that uses (again) consensus to approach the internal model of the exosystem and thereby achieves perfect output regulation. Our approach leverages methods from internal model based controller synthesis, multi-agent consensus over directed networks, and stability of time-varying linear systems; the derived result is an adaptation of the (centralized) internal model principle to the distributed, networked setting.

I Introduction

Over the past decade, many distributed control problems of networked multi-agent systems have been extensively studied; these include e.g. consensus, averaging, synchronization, coverage, and formation (e.g. [1, 2, 3]). Progressing beyond first/second-order and homogeneous agent dynamics, the distributed output regulation problem with general linear (time-invariant, finite-dimensional) and heterogeneous agent dynamics has received much recent attention (e.g. [4, 5, 6, 7, 8, 9]). In this problem, a network of agents each tries to match its output with a reference signal, under the constraint that only a few agents can directly measure the reference. The reference signal itself is typically generated by an external (linear) dynamic system, called “exosystem”. The distributed output regulation problem not only subsumes some earlier problems like (leader-following) consensus and synchronization, but also addresses issues of disturbance rejection and robustness to parameter uncertainty. Also see e.g. [10, 11] for further extensions of this problem to deal with nonlinear agent dynamics.

Output regulation has a well-studied centralized version: A single plant tries to match its output with a reference signal (while maintaining the plant’s internal stability) [12, 13, 14]. In the absence of system parameter uncertainty, the solution of the “regulator equations”, embedding a copy of the exosystem dynamics, provides a solution to output regulation [14]. When system parameters are subject to uncertainty, a dynamic compensator/controller must be used embedding -copy of the exosystem, where is the number of (independent) output variables to be regulated. The latter is well-known as the internal model principle [13]. These methods for solving the centralized output regulation problem, however, cannot be applied directly to the distributed version, inasmuch as not all agents have direct access to the reference signal or the exosystem dynamics.

The distributed output regulation of networks of heterogeneous linear agents is studied in [5]. The proposed distributed controller consists of two parts: an exosystem generator and a controller based on regulator equation solutions. Specifically, the exosystem generator of each agent aims to (asymptotically) synchronize with the exosystem using consensus protocols, thereby creating a local copy of the exosystem. Meanwhile each agent independently tracks the signal of its local generator, by applying standard centralized methods (in [5] regulator equation solutions). This solution effectively separates the controller synthesis into two parts – distributed exosystem generators by network consensus and local output regulation by regulator equation solution.

One important limitation, however, of the above solution is: in both the exosystem generator design and the regulator equation solution, it is assumed that each agent uses exactly the same dynamic model as the exosystem. This assumption may be unreasonable in the distributed network setting, because those agents that cannot directly measure the reference signal are unlikely to know the precise dynamic model of the exosystem. To deal with this challenge, [8] proposes (in the case of static networks) an “adpative” exosystem generator and an adaptive solution to the regulator equations. In essence, each agent runs an additional consensus algorithm to update their “local estimates” of the exosystem dynamics.

All the regulator-equation based solutions above fall short in addressing the issue of system parameter uncertainty. In practice one may not have precise knowledge of some entries of the system matrices, or over time the values of some parameters drift. The distributed output regulation problem considering parameter uncertainty is studied in [4, 6]. The proposed controller is based on the internal model principle, but does not employ the two-part structure mentioned above. It seems to be for this reason that restrictive conditions (acyclic graph or homogeneous nominal agent dynamics) have to be imposed in order to ensure solving output regulation. Moreover, it is also assumed in [4, 6] that each agent knows the exact model of the exosystem dynamics.

In this paper, we provide a new solution to the distributed output regulation problem of heterogeneous linear systems (generally non-minimum phase), where the agents do not have an accurate dynamic model of the exosystem and the agent dynamics are subject to parameter uncertainty. In particular, we propose to use the two-part structure of the distributed controller in the following manner: The first part is an exosystem generator that works over time-varying networks ([15, 16])111This was apparently developed in [15] and in [16] independently. The first versions of [15] and [16] appeared on arXiv.org, with the former three months earlier than the latter. We thank Dr. Liu and Dr. Huang for in a correspondance bringing our attention to their work., and the second part is a dynamic compensator embedding an internal model of the exosystem that addresses parameter uncertainty. The challenge here is, in the design of the dynamic compensator, those agents that cannot directly measure the exosystem have no knowledge of the internal model of the exosystem; on the other hand, we know from [13] that a precise internal model is necessary to achieve perfect regulation with uncertain parameters. To deal with this problem, we propose an extra consensus protocol to update the agents’ local estimates of the internal model of the exosystem; in this process, a unique feature is to avoid certain transmission zeros of the agents’ dynamics in order to guarantee the existence of a dynamic compensator. Inasmuch as the agents gradually ‘learn’ the internal model of the exosystem in a purely distributed fashion, we call this control design ‘distributed internal model principle’ (with reference to the centralized version in [13]).

The main contribution of this paper is the proposed internal-model based distributed controller, with the novel design of a dynamic compensator that does not require all agents to know the internal model (-copy) of the exosystem a priori. This proposed controller provably solves the distributed output regulation problem in which the following constraints/conditions simultaneously hold:

• Initially unknown dynamic model of the exosystem. This is not considered in [4, 6].

• Parameter uncertainty of agent dynamics. This is not addressed in [5, 8].

• Non-minimum phase agent dynamics. This is not dealt with in [7].

• Time-varying directed networks. This is not addressed in [8, 9, 10, 11].

• Heterogeneous agent dynamics. This is not dealt with in [9].

In addition we note that [15] proposes a distributed controller to solve the consensus problem whose design idea is similar to ours. We point out, however, a few important differences. First, the consensus problem is different from the output regulation problem (the former is usually viewed as a special case of the latter with full-state observation). Second, while [15] deals with a class of nonlinear systems, the eigenvalues of the exosystem are required to be distinct. We do not make such an assumption; thus (i) the set of signals that can be generated by the exosystem is a strict superset of that in [15], and (ii) the minimal polynomial of the exosystem is generally different from the characteristic polynomial. Third, our designed distributed controller is based on the internal model principle, which is different from the controller in [15]. Finally, while the parameter uncertainty considered in [15] is represented by a vector, the uncertainty in this paper is represented by matrices.

The paper is organized as follows. Section II introduces the concept of communication graphs and formulates the distributed output regulation problem. Section III presents the solution distributed controller, and Section IV states our main result. In Section V We design a more general distributed controller. Section VI illustrates our results by a simulation example. Finally, Section VII states our conclusions.

Ii Preliminaries

In this paper, we will use the following notation. Let , and be the identity matrix. For a complex number , denote its complex conjugate by . Write for the closed right half (complex) plane; for the set of all eigenvalues of . We say that a (square) matrix is stable if the real parts of its eigenvalues are negative.

Ii-a Agents and an exosystem

We consider a network of agents that are linear, time-invariant, and finite-dimensional. The dynamics of each agent is given by

 ˙xi=Aixi+Biui+Piw0 (1) zi=Cixi+Diui+Qiw0 (2)

where is the state vector, the control input, the output to be regulated, and the exogeneous signal generated by the exosystem

 ˙w0=S0w0. (3)

Here and are real matrices of appropriate sizes. The signal represents reference to be tracked and/or disturbance to be rejected: in (1) represents disturbance acting on the agent ’s dynamics and in (2) represents reference signals to be tracked by agent .

Note that the agents are generally heterogeneous: Each of the matrices and may have different dimensions and entries. Furthermore, we consider that the matricies may have uncertainty; namely

 Ai=Ai0+ΔAi, Bi=Bi0+ΔBi, Ci=Ci0+ΔCi, Di=Di0+ΔDi, Pi=Pi0+ΔPi, Qi=Qi0+ΔQi (4)

where are the nominal parts of agent and are the uncertain parts.

Ii-B Communication digraphs

Given a multi-agent system with agents and an exosystem, we represent the time-varying interconnection among the agents and the exosystem by a digraph , where , , is the node set, and is the edge set. The node , , represents the th agent, and the node 0 the exosystem. Moreover, is the node set including the exosystem and is the node set except for the exosyetm. The th node receives information from the th node at time if and only if . Then the union digraph for a time interval is defined as .

Definition

The digraph uniformly contains a spanning tree if there is such that for every the union digraph contains a spanning tree.

We define the communication weight by if and if . We assume that is piecewise continuous and bounded for all (a technical assumption to be used in Lemma IV below). Note that the exosystem does not receive information from all the agents for all , and thus .

For time and digraph , the graph Laplacian is defined as

 lij(t):={∑Nj=0aij(t),i=j−aij(t),i≠j

where .

Ii-C Problem statement

We represent by the time-varying interconnection among the agents and the exosystem again. We regard the exosystem as node 0. In particular, at any time only some agents (possibly different across time) can receive information from the exosystem. This differs the current problem from the traditional, centralized output regulation problem [12, 17, 13, 14].

Problem (Distributed Output Regulation Problem)

Given a network of agents (1), (2), (4) and an exosystem (3) with interconnection represented by , design for each agent a distributed controller such that

1. if for all , then as for all and all (internal stability); and

2. as for all and all (output regulation).

Ii-D Example

The literature [8] treat almost the same problem. The literature uses the regulator equations approach. This approach is successful to address no-perturbation case. In this section, we show the example that the algorithm in [8] cannot solve a distributed output regulation ploblem if there are perturbation terms. We consider that the exosystem has , and 3 agents have and for all . We also consider the (time-invariant) network as displayed in Fig. 2. The node numbered 0 is represented the exosystem. Then the initial states are selected uniformly at random from the interval .

The simulation results of [8] is shown in Fig. 2. If there is perturbation, . Therefore we need to design the controller that ensures even if there are parameter uncertainties. In order to address the robust distributed output regulation problem, it is necessary to use an internal-model approach.

Iii Structure of Distributed Controller

At the outset we make the following assumptions.

Assumption

The digraph uniformly contains a spanning tree and its root is node 0 (the exosystem).

Assumption

For each agent , is stabilizable.

Assumption

For each agent , is detectable.

Assumption

For each agent and for every eigenvalues of ,

 rank[Ai0−λIniBi0Ci0Di0]=ni+qi. (5)

Assumption

The real parts of all eigenvalues of are zeros.

Remark

Assumption III and Assumptions III-III are necessary conditions for consensus over time-varying networks [18] and for output regulation [12], respectively. Only Assumption III is a sufficient condition for output regulation; the reason is the following. By [19, 3.4. Discussion], if contains exponentially unstable modes, then one needs to consider stronger connectedness assumptions on the digraph . The exogeneous signal with in the unstable mode diverge exponentially fast. In order to track the diverging signal, the agents must be connected to the exosystem with sufficient large weights. It is difficult to make assumptions satisfying the conditions that the agents can track the exogeneous signal. In this work, we make Assumption III for simplicity.

Remark

By [12], Assumption III means that the transmission zeros of agent are disjoint from all eigenvalues of , and also implies that the number of outputs be no more than that of inputs, i.e. .

Because not all agents can access the exosystem (i.e. cannot be measured by all agents), we cannot use (2) directly. Instead we consider the following (estimated) error vector

 ei=Cixi+Diui+Qiwi∈Rqi (6)

where is the estimated exogeneous signal. This is in (2) with replace by .

In order to solve Problem II-C, we describe two parts of the structure of the controller: (1) distributed exosystem generator and (2) distributed dynamic compensator.

Iii-a Distributed exosystem generator

It is reasonable that each agent has a local estimate of the exosystem’s dynamics since not all agents can access the exosystem. Let be the estimete of and consider

 ˙Si(t)=N∑j=0aij(t)(Sj(t)−Si(t)), (7) ˙wi(t)=Si(t)wi(t)+N∑j=0aij(t)(wj(t)−wi(t)). (8)

By using (7) and (8), it is guaranteed under Assumption III that

 limt→∞(Si(t)−S0)=0,limt→∞(wi(t)−w0(t))=0.

We show this statement in detail in Section IV below.

This protocol is used to approximate the exosystem for each agent . Thus we call (7) and (8) the exosystem generator.

Equations (7) and (8) has also been used in [15] for the adaptive distributed observer, and first proposed in [8] but for time-invariant networks.

Iii-B Distributed dynamic compensator

We consider the following dynamic compensator

 ˙ξi=Ei(t)ξi+Fi(t)ei ui=Ki(t)ξi (9)

where is defined in (6). Our strategy is the following:

1. We obtain by using (7) and (8), therefore .

2. Design and such that .

3. The desired will ensure.

In order to specify the matrices in (9), we extend the control law in [20, Section 1.3] to the multi-agent system setting. Let be the roots of the minimal polynomial of . Note that . Then we define . Let , , be the coefficients of the polynomial satisfying

 sk+c0,1(λ0)sk−1+⋯+ c0,k−1(λ0)s+c0,k(λ0) =k∏d=1(s−λ0,d(t)). (10)

For each agent , let be a local estimate of , and , , the estimated coefficients generated by that satisfy

 sk+ci,1(λi)sk−1+⋯+ ci,k−1(λi)s+ci,k(λi) =k∏d=1(s−λi,d(t)). (11)

Consider the following consensus algorithm:

 ˙λi(t)=N∑j=0aij(t)(λj(t)−λi(t)), λi(0)∈jRk. (12)

It follows from Assumption III that as . As a result, the coefficient for each . Note that by Assumption III the entries of are purely imaginary, and hence we only need to consider the initial condition (thus for all ).

Since we consider that the agents may have uncertainty, the regulator equation approach (e.g. [8]) does not work (as shown in Fig. 2). For the robust output regulation problem, we consider the -copy internal model as [20, Section 1.3]. Let be the -copy internal model, where

 H′i:=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣00⋮01⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦. (13)

We state the following lemma using the above matrices.

Lemma

Assume that Assumptions IIIIII hold. Then the following pair of matrices

 ([Ai00HiCi0Gi(λi)],[Bi0HiDi0])

is stabilizable if

 rank[Ai0−λi,dIniBi0Ci0Di0]=ni+qi (14)

for each .

Proof

Refer to the proof of [20, Lemma 1.26] with the condition (14) and the fact that in (12) and thus for all .

In Lemma III-B the sufficient condition (14) means that every does not correspond to the transmission zeros of agent . In (14), is time-varying because it is updated according to (12). Since for all , if agent ’s dynamics has purely imaginary transmission zeros, it is possible that (14) is violated. In order to satisfy (14) for all , we make the following (simpifying) assumption.

Assumption

For every agent , there are no transmission zeros on the imaginary axis, i.e.

 rank[Ai0−λIniBi0Ci0Di0]=ni+qi

for all .

If every agent is minimum-phase, then Assumption III-B is satisfied. In addition, this assumption allows transmission zeros on the open right (complex) plane, thus admitting non-minimum-phase system. In the case where Assumption III-B does not hold, it is a challenge to ensure that (14) holds for all . Nevertheless, in Section V below we shall present a novel strategy to guarantee (14) even in the presence of purely imaginary transmission zeros.

From Lemma III-B and Assumption III-B, we may synthesize such that the matrix

 [Ai00HiCi0Gi(λi)]+[Bi0HiDi0][Ki1(λi) Ki2(λi)] (15)

is stable for all . In addition, we choose such that the matrix is stable under Assumption III.

Now we are ready to present the matrices and in the dynamic compensator (9):

 Ei(λi):= [Ai0−LiCi000Gi(λi)]+[Bi0−LiDi00]Ki(λi), Fi:=[LiHi], Ki(λi):=[Ki1(λi) Ki2(λi)]. (16)

Note that in (16), and are time-varying as is time-varying, while is time-invariant; and by (12) there hold

 Gi(λi)→Gi(λ0) Ki(λi)→Ki(λ0) Ei(λi)→Ei(λ0).

Using the distributed dynamic compesator (9), we ensure that the estimated error and the output to be regulated converge to 0. For details, see next section.

Iv Main Result

Our main result is the following.

Theorem

Given the multi-agent system (1), (2), (4) and the exosystem (3), suppose that Assumptions III-III-B hold and , , , , , are sufficiently small. Then for each agent , the distributed exosystem generator (7), (8) and the distributed dynamic compensator (9) with (12), (16) solve Problem II-C.

Several remarks on Theorem IV are in order.

Remark

For the distributed output regulation problem, Theorem IV extends previous results in the literature in several aspects: The proposed distributed controller (i) employs (time-varying) internal model to deal with system parameter uncertainty (cf. [21, 5, 8]), (ii) needs no a priori knowledge of the exosystem (cf. [21, 4, 5, 6]), and (iii) deals with generally non-minimum-phase systems (cf. [7]).

Remark

This algorithm is useful even for the single-agent system. In this case, our approach is the extension of the conventional method [12, 17, 13, 14][20, Section 1.3]. Even if the agent does not know the exosystem at first, the output regulation problem is solvable by the (disributed) exosystem generator and the (distributed) dynamic compensator.

Remark

If the exosystem is a leader agent that possesses computation and communication abilities, then the leader can compute the roots of its own minimal polynomial and send the information to other connected agents. If the exosystem is some entity that cannot compute or communicate, then those agents that can measure the exosystem (in particular know ) compute the corresponding minimal polynomial and the roots, and send the information to the rest of the network.

Remark

For each agent to “learn” the internal model of the exosystem, our strategy is to make the agents reach consensus by (12) for the roots of the exosystem’s minimal polynomial (i.e. eigenvalues of ). It might appear more straightforward to reach consensus for the coefficients of the exosystem’s minimal polynomial; the advantage of updating with (12), nevertheless, is that we may directly guarantee the equality in (5) in Assumption III.

Remark

In (7), there are entries to update and communicate. If the minimal polynomial of equals its characteristic polynomial () and is in the companion form

where , , and are as defined in (10), then each agent does not need to exchange and update the whole . Each agent only needs to exchange and update by (12) and make also in the companion form.

Remark

In the equation (12), we do not need to use all entries of , because the eigenvalues of the real matrices must be in conjugate pairs. Indeed, for all we may write in the following form

 λi(t)=⎧⎪ ⎪⎨⎪ ⎪⎩[^λi(t)⊤ ^λ∗i(t)⊤]⊤,k is an even number[^λi(t)⊤ ^λ∗i(t)⊤ 0]⊤,k is an odd number

where . From this form, each agent can make their entire after exchanging and updating only .

To prove Theorem IV, we need the following two lemmas. Their proofs are presented in [15].

The first lemma states a stability result for a particular type of time-varying systems. This lemma is generalized version of [15, Proof of Lemma 3.1].

Lemma

Consider

 ˙x(t)=A1(t)x(t)+A2(t)x(t)+A3(t) (17)

where are piecewise continuous and bounded on . Suppose that the origin is a uniformly exponentially stable equilibrium of , and , as . Then as .

The second lemma asserts that the distributed exosystem generators proposed in Section III-A synchronize with the exosystem. This lemma is the same as [15, Lemma 3.1].

Lemma

Consider the distributed exosystem generator (7) and (8). If Assumption III holds, then

 limt→∞(Si(t)−S0)=0,limt→∞(wi(t)−w0)=0

for all .

Now we are ready to prove Theorem IV.

Proof of Theorem IV: Let be the combined state. From (1), (2), (6) and (16), we derive

 ˙ηi =Mi(λi)ηi+[0FiQi]wi+[Pi0]w0 (18) zi =[Ci DiKi(λi)]ηi+Qiw0. (19)

where

 Mi(λi):=[AiBiKi(λi)FiCiEi(λi)+FiDiKi(λi)]= \scalebox0.75$⎡⎢⎣AiBiKi1(λi)BiKi2(λi)LiCiAi0−LiCi0+(Bi0+LiΔDi)Ki1(λi)(Bi0+LiΔDi)Ki2(λi)HiCiHiDiKi1(λi)Gi(λi)+HiDiKi2(λi)⎤⎥⎦$. (20)

First, we define

 T:=⎡⎢⎣I0000I−II0⎤⎥⎦

and obtain

 TMi(λi)T−1= \scalebox0.8$⎡⎢ ⎢⎣Ai+BiKi1(λi)BiKi2(λi)BiKi1(λi)HiCi+HiDiKi1(λi)Gi(λi)+HiDiKi2(λi)HiDiKi1(λi)Δ1Δ2Ai0−LiCi0+Δ3⎤⎥ ⎥⎦$,

where

 Δ1 =−ΔAi+LΔCi−(ΔBi−LΔDi)Ki1(λi) Δ2 =−(ΔBi−LΔDi)Ki2(λi) Δ3 =−(ΔBi−LΔDi)Ki1(λi).

If are all zeros, the upper-left submatrix equals (15) and thus the submatrix is stable. Also is stable. Since and are similar, is stable for all . Even if there exist sufficiently small perturbation terms, from the continuity of eigenvalues (with respect to matrix entries), remains stable for all . This implies that internal stability (the (i) in Problem II-C) is satisfied.

Next, from Assumption III and the above statement we have , and thus the following equations

 Xi(λ0)S0 =Mi(λ0)Xi(λ0)+[PiFiQi] 0 =[Ci DiKi(λ0)]Xi(λ0)+Qi (21)

have a unique solution (see e.g. [22, Appendix A]). Let , . Then from (18) and (21) we obtain

 \scalebox0.9$+[Pi0]w0−Xi(λ0)S0w0$ \scalebox0.9$+(Mi(λ0)Xi(λ0)+[PiFiQi]−Xi(λ0)S0)w0$ =\scalebox0.9$(Mi(λ0)~ηi)+(~Mi~ηi)$ \scalebox0.9$+(~MiXi(λ0)w0+[0FiQi](wi−w0))$.

From Lemma IV, . Moreover, is stable and . Therefore, from Lemma IV.

Furthermore from (19) and (21) we obtain

 zi =[Ci DiKi(λi)](~ηi+Xi(λ0)w0)+Qiw0 =[Ci DiKi(λi)]~ηi+([Ci DiKi(λi)]Xi(λ0)+Qi)w0.

Since and

 [Ci DiKi(λi)]Xi(λ0)+Qi →[Ci DiKi(λ0)]Xi(λ0)+Qi=0

we conclude that as (the (ii) in Problem II-C).  \QED

Remark

In Theorem IV, the requirement that , , , , and be sufficiently small can be relaxed. Indeed from the proof above, , , and need not be small as long as the matrix in (20) remain stable; and can be arbitrary.

V Purely Imaginary Transmission Zeros

In this section, we generalize Theorem IV by designing a distributed controller for the case where Assumption III-B does not hold, i.e. there exist transmisssion zeros of the agents on the imaginary axis. In this case, because each (vector) is updated continuously, entries of may coincide with the transmission zeros of agent , which would violate (14). Consequently we cannot design with Lemma III-B.

In order to choose the local estimate satisfying the condition (14) and design , must converge to and at the same time avoid the transmission zeros of the agent . Fig. 3 shows examples of the trajectory of . The circles and the crosses represent respectively the transmission zeros of the agent and the eigenvalue of .

The initial value is in and moves toward . We divide the arrangement of transmission zeros into three cases:

1. If there is no purely imaginary transmission zero of agent (see Fig. 3(i)), then Assumption III-B holds and we need no further control design.

2. If there is a purely imaginary transmission zero of agent , and moves close to it (see Fig. 3(ii)), then should move in a semicircle dented to the right around the transmission zero. By moving to the right, is always in and thus Lemma III-B is guaranteed for all .

3. If there is a purely imaginary transmission zero of agent , and there are also other transmission zeros on the open right-half-plane (see Fig. 3(iii)), the radius of semicircle should be smaller than (e.g. half of) the distance between these transmission zeros.

To formalize the above idea, we define several quantities. Let

 Πi:={s∈C+ ∣∣∣ rank[Ai0−sIniBi0Ci0Di0]

be the set of closed right-half-plane transmission zeros of agent and

 ~Πi:={s∈Πi ∣∣ Re(s)=0} (23)

the subset of purely imaginary transmission zeros. We do not need to avoid open left-half-plane transmission zeros because . Note that Assumption III and are equivalent, and Assumption III-B holds if and only if .

We define a new function. For two sets of finite number of complex numbers, define the distance between and by

 dist(S1,S2):=min{|s1−s2| ∣∣ s1∈S1,s2∈S2}.

Then define the radius of the semicircle shown in Fig. 3 as

 ρi:= (24)

This has three cases:

1. If there is no purely imaginary transmission zeros of agent , i.e. , then Assumption III-B holds and simply the radius is zero.

2. If there are only purely imaginary transmission zeros of agent , i.e. , then the radius is half of the distance between and .

3. If there are transmission zeros of agent on both purely imaginary axis and right-half-plane, then the radius is half of the smaller of the distance between and and that between and .

In the definition of , we consider the coefficient for simplicity, but we can choose any coefficient from . By using this , we ensure the radius of the semicircle for the three cases as illustrated in Fig. 3.

Then we consider for all ,

 λi(t)=αi(t)+jβi(t)∈Ck, αi(t),βi(t)∈Rk (25) ˙βi(t)=N∑j=0aij(t)(βj(t)−βi(t)) (26) αi,d(t)=⎧⎨⎩0,γi,d(t)≥ρi or i=0√ρ2i−γ2i,d(t),γi,d(t)<ρi and i≠0 (27)

where

 d=1,…,k αi(t)=[αi,1(t) ⋯ αi,k(t)]⊤ βi(t)=[βi,1(t) ⋯ βi,k(t)]⊤ γi,d(t):={0,~Πi=∅dist({jβi,d(t)},~Πi),otherwise.

Note that means the distance between and its closest (purely imaginary) transmission zero, and for all . Moreover from Assumption III, for all . Finally, we update by (26) and (27) instead of (12). It follows immediately from these definitions the following result.

Lemma

Consider the equations (25), (26), (27). If Assumption III holds, then , , converge to while avoiding transmission zeros of the agent .

Using the above method, we state the main result of this section.

Theorem

Given the multi-agent system (1), (2), (4) and the exosystem (3), suppose that Assumptions III-III hold and , , , , , are sufficiently small. Then for each agent , the distributed exosystem generator (7), (8) and the distributed dynamic compensator (9) with (16), (25), (26), (27) solve Problem II-C.

Proof

From Lemma IV, we have and . From Lemmas III-B and V, each satisfies the condition (14) and thus we can define for all . The remainder of the proof is identical to the proof of Theorem IV.

Remark

As in Remark IV, we do not need to use all entries of